MinT - Best Known (59−30, 59, s)-Nets in Base 64

Best Known (59−30, 59, s)-Nets in Base 64

(59−30, 59, 513)-Net over F₆₄ — Constructive and digital

Digital (29, 59, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 59, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(59−30, 59, 1288)-Net over F₆₄ — Digital

Digital (29, 59, 1288)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁵⁹, 1288, F₆₄, 3, 30) (dual of [(1288, 3), 3805, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(64⁵⁹, 1366, F₆₄, 3, 30) (dual of [(1366, 3), 4039, 31]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(64⁵⁹, 4098, F₆₄, 30) (dual of [4098, 4039, 31]-code), using
    - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
      1. linear OA(64⁵⁹, 4096, F₆₄, 30) (dual of [4096, 4037, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
      2. linear OA(64⁵⁷, 4096, F₆₄, 29) (dual of [4096, 4039, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(59−30, 59, 1296377)-Net in Base 64 — Upper bound on s

There is no (29, 59, 1296378)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 36696 089552 006804 378231 138060 208233 707680 031137 642145 803595 984511 599800 781016 673490 221784 533576 468610 532344 > 64⁵⁹ [i]